Almost Everywhere Convergence of the inverse Jacobi Transform and Endpoint Results for a Disc Multiplier

TL;DR

This paper studies the convergence of the inverse Jacobi transform and introduces a maximal operator for the disc multiplier in Jacobi analysis, providing new endpoint results and mapping properties.

Contribution

It introduces a maximal operator for the Jacobi disc multiplier and establishes its mapping properties, including endpoint weak type results, advancing understanding of convergence in Jacobi analysis.

Findings

Established weak type endpoint results for the maximal operator

Determined mapping properties of the maximal operator

Provided sharp almost everywhere convergence results for the inverse Jacobi transform

Abstract

We introduce a maximal operator for the natural analogue of the disc multiplier in the framework of Jacobi analysis and determine the mapping properties of said maximal operator, including weak type endpoint results. In particular we obtain sharp information on the almost everywhere convergence of the inverse Jacobi transform.